- Computer Algebra Software for Polynomial Control Systems

Polycon

- Computer Algebra Software for Polynomial Control Systems

Krister Forsman

Dept. of Electrical Engineering, Linkoping University S-581 83 Linkoping, Sweden

email:

krister@isy.liu.se

1993-02-15

REGLERTEKNIK

AUTOMATIC CONTROL LINKÖPING

Technical reports from the automatic control group in Linkoping are available by anonymous ftp at the address

130.236.24.1 (joakim.isy.liu.se)

This report is contained in the compressed ps-le named

/pub/reports/LiTH-ISY-R-1447.ps.Z

Polycon

- Computer Algebra Software for Polynomial Control Systems

Krister Forsman

Dept. of Electrical Engineering

Linkoping University, S-581 83 Linkoping, Sweden email:

krister@isy.liu.se

1993-02-15

Abstract.

This paper describes the features and implementation of the Maple package

^Polycon

which is intended to assist the control theorist in the analysis of nonlinear dynamical systems, in continuous and discrete time.

^Polycon

handles systems where all nonlinearities are polynomial or rational functions. It implements functions that are not available at a \usable" level in other programs, to the author's knowledge. It is supposed to be accessible to non-experts and those that are not familiar with computer algebra, commutative algebra or dierential algebra.

^Polycon

is included in the Maple Share Library and thus available by anonymous ftp.

Keywords:

computer algebra, symbolic manipulation, CACSD, polynomial control systems, state space analysis, external behavior, Gr obner bases, commutative algebra, dierential algebra

1 Introduction

When discussing how constructive or how algorithmic a theory is, the following levels provide a rough classication:

abstract meaning that e.g. existence of some objects is proved, but it is not clear how to construct those objects or if this is even possible.

eective algorithms for constructing the objects discussed are given,

even though these algorithms may be of small value for practical purposes, due to high complexity

ecient ecient algorithms for constructing the objects involved are given

there is a computer program that somebody knows how to run that computes the desired entities

usable there is a computer program for computing the desired objects that is not machine dependent and that other people

than its author(s) are able to run.

The dierence between ecient and eective is of course depending on many circumstances, such as available hardware etc, so it is often very dicult to tell which case is at hand.

If we look at the development of commutative, dierential and dierence algebra in the

theory of nonlinear control systems we may discern approximately the following features:

abstract Much of the work by M. Fliess 9, 10, 8]

some work by E.D. Sontag 31], T. Glad and others

eective/ecient Most of the work by Glad 22, 23, 26] and S. Diop 5, 6], some of the work by Fliess.

The word abstract does denitely not have a negative meaning in this context. A large part (maybe the majority) of nonlinear control systems research today would probably be classied as abstract with the denitions above.

The

^Polycon

package, described in this paper, is intended to be one step in the direction from ecient to usable. Its theoretical basis is commutative algebra rather than dierential algebra, the reason being that the constructive aspects of commutative algebra are better understood than those of dierential algebra. There is a large amount of research going on in the area of constructive commutative algebra both from a mathematical and a computer science point of view, whereas only a limited number of persons are developing the algorithmic aspects of dierential algebra, unfortunately.

Maple 3] is probably the most widespread symbolic algebra program today, along with Mathematica. It has been available for more than ten years and its engineering applica- tions abound. Some examples of Maple software for dealing with nonlinear control sys- tems are 4, 32, 34]. The package

^Polycon

is included in the Maple Share Library and is thus available by anonymous ftp at the addresses

129.132.101.33 (neptune.inf.ethz.ch)

and

129.97.140.58 (daisy.waterloo.edu)

. It is also possible to get it anonymously from Linkoping at the address

130.236.24.1 (joakim.isy.liu.se)

, under the directory

/pub/src/maple/polycon

. The size of the le containing all the Maple source code for

Polycon

, including help texts, is 85 kbytes.

Polycon

is a collection of Maple procedures for the analysis of polynomial and rational dynamical systems written either in state space form, i.e. in the form

_

x

(

^t

) =

^f

(

^x

(

^t

)

^u

(

^t

))

^{ y}

=

^h

(

^xu

) (1) or

x

(

^t

+ 1) =

^f

(

^x

(

^t

)

^u

(

^t

))

^{ y}

=

^h

(

^xu

) (2) where

^h

and all components of

^f

are rational functions in

^x

and

^u

, or in input-output form, i.e.

p

(

^{y0:::ynu0:::ur}

) = 0 (3) where subindices denote either time-derivatives or time-shifts and

^p

is a polynomial. Most functions are available in a continuous time and a discrete time version the convention is that the names of continuous time functions end in

^c

and those of discrete time functions end in

^d

. Most functions handle systems with several inputs.

The package is aimed at questions related to state space realizations, such as

retrieving external behavior from a state space description, i.e. \conversion" from a state space description ((1) or (2)) to an input-output description (3):

ss2ioc, ss2iod

realization of input-output equations:

par2ssc, io2ssc, io2ssd

(algebraic) observability, as dened in e.g. 19, 23]:

obsvc, obsvd

state transformations of systems:

newsysc, newsysd

nding the transformation between two i/o-equivalent systems:

sstrac, sstrad

There is also a procedure

^loclyap

for analysis of local Lyapunov functions.

Note that there are many topics in control theory that are not represented at all in

Polycon

, such as controllability, optimal control and feedback. The reasons for this is that there are basically no algorithms general enough for solving these problems in the algebraic framework, to the author's knowledge. One can say that the emphasis is on analysis rather than design, mainly because analysis is simpler

^:::

This paper is organized as follows:

In section 2 we give a user-level description of the most important

^Polycon

functions.

Section 3 very briey discusses some of the mathematical background and the algo- rithms used, mainly by referring to earlier work.

In section 4 we show how

^Polycon

works on a simple example.

Section 5 contains an outline of the possible extensions of the procedures and the functionality of

^Polycon

.

2 The Main Functions

The functions available in

^Polycon

are, in alphabetical order:

auxlieder io2ssc io2ssd lieder liederlist liehom liehomlist loclyap newsysc newsysd obsvc obsvd par2ssc ss2ioc ss2iod sstrac sstrad uyder uyhom xss2ioc xss2iod

All functions have detailed help texts. For more information about a particular function, use the help command

^?

after having loaded

^Polycon

, e.g.

^?sstrac

You can also type

^?polycon

to obtain general information on

^Polycon

.

In

^Polycon

time derivatives and time shifts of dependent variables are represented by subindices, as done by e.g. Ritt 30]. This means that e.g. in continuous time

y

0

=

^y

(

^t

)

^{ y1}

= _

^y

(

^t

)

^{ y2}

= 

^y

(

^t

)

^

etc

^:

(4) while in discrete time

y

0

=

^y

(

^t

)

^{ y1}

=

^y

(

^t

+ 1)

^{ y2}

=

^y

(

^t

+ 2)

^

etc

^:

(5) This rule holds for input and output variables. Higher derivatives or shifts of state variables do not occur, so state variables are still called

^x1:::xn

by default. If two state descriptions are involved the second one has state variables

^z1:::zn

by default. In functions where the

rst derivative or shift of the state variables may occur, e.g.

^io2ss

and

^newsys

, they are

denoted by

^dx1:::dxn

. In Maple, subscript is represented by concatenation so that e.g.

^y2

is written

^y2

.

'

i

(

^xiuy

) = 0

^-



io2ssc

obsvc ^x

_ =

^f

(

^xu

)

y

=

^h

(

^xu

)

-

ss2ioc



io2ssc, par2ssc ^p

(

^uy

) = 0

_

z

=

^g

(

^zu

)

y

=

^j

(

^zu

)

? 6

sstrac, newsysc

Figure 1: Diagram describing continuous time functions.

'

i

(

^xiuy

) = 0

^-



io2ssd obsvd

x

+

=

^f

(

^xu

)

y

=

^h

(

^xu

)

-

ss2iod



io2ssd ^p

(

^uy

) = 0

z

+

=

^g

(

^zu

)

y

=

^j

(

^zu

)

? 6

sstrad, newsysd

Figure 2: Diagram describing discrete time functions.

The largest part of the package is aimed at questions related to rational state realizations of control systems. The diagram in gure 1 summarizes the use of the dierent procedures in the continuous time case. In the diagram

^p

and

^'i

are dierential polynomials in

^uy

, i.e. they involve time derivatives of

^u

and

^y

(but not of

^xi

).

For discrete time systems the functions in gure 2 are available. Here

^p

and

^'i

involve time shifts of

^u

and

^y

(but not of

^xi

).

The functions

^ss2ioc

and

^ss2iod

take as input a system in state space form, rep- resented by the polynomial rhs vector eld

^f

(an object of type

^vector

in Maple) and the output-map

^h

(a polynomial) and return a polynomial in the input, the output and their derivatives representing the input-output behavior of the system.

These two procedures work for many rational systems as well, i.e. systems of the type (1) or (2) where

^h

and all

^fi

are rational functions of their arguments. However, there are examples where the algorithm fails, see e.g. 13].

The functions

^obsvc

and

^obsvd

take as input a system in state space form and one

of the state variables,

^xi

say, and return two polynomials in the input, the output and their

derivatives and

^xi

constituting an observer relation for

^xi

. Such a relation exists i the system is algebraically observable, i.e. the input-output equation is of order equal to the state space dimension.

The user can choose if he only wants to accept as an answer a list of rational expressions for

^xi

in inputs and output (and their derivatives) or if he allows

^xi

to occur nonlinearly. In the latter case the answer is a list of polynomials

^'i

in inputs and output and

^xi

such that

'

i

= 0 under the system in question.

The functions

^io2ssc

and

^io2ssd

can be thought of as the inverses of

^obsvc

and

^obsvd

. The input arguments are a polynomial input-output equation (represented by a polynomial

p

only) and a list of rational expressions for state variable candidates, i.e. a list

^S

of rational functions in

^uy

and their derivatives such that the

ⁱ

:th element of

^S

might serve as the

ⁱ

:th state. The function returns a list consisting of the rhs and the output map of the state space equation for the continuous time polynomial SISO system given by the i/o-equation

^p

= 0.

The procedure

^par2ssc

returns the rhs vector of the state equation for a continuous time rational SISO system given in i/o-form. Its input is a list

^H

of rational functions that parametrize the hypersurface

^p

= 0, where

^p

is the i/o-relation.

There is no function

^par2ssd

, because it is not clear at the moment whether it is al- ways possible to go from a rational parametrization of a hypersurface to a realization of the corresponding discrete time system.

The procedures

^sstrac

and

^sstrad

suppose that the two systems involved are i/o- equivalent and then nd the state space transformation between them.

The functions

^newsysc

and

^newsysd

take a system in state space form and a transfor- mation as inputs and returns the system in the new coordinates, dened by the transforma- tion.

For a full description of all

^Polycon

functions we refer either to the online help texts or to the technical report 15] which is available by anonymous ftp. This report contains exactly the same information as is available online in Maple plus some source code.

3 The Implementation

Let us here very briey mention something about the mathematics behind

^Polycon

.

Polycon

heavily relies on elimination theory for commutative rings and in particular Grobner bases (GB) computations. For an introduction to the theory of GB see e.g. 2, 18, 21, 29]. All major computer algebra programs (Maple, Mathematica, Reduce, Axiom and Macsyma) have GB packages of varying depth and quality. The branch of mathematics dealing with e.g. GB is currently very active and a lot of progress is still made in this research area.

Grobner bases can be seen as the generalization of Gaussian elimination to systems of

polynomial equations. A GB is a generating set for a polynomial ideal having some appealing

properties. Most important in this context is the following elimination property, which we

state without giving any details, since that would lead much too far:

Let

^a

be an ideal in

^k



^{X1 :::Xn}

] and partition

^X1:::Xn

into two disjoint sets

^S

and

^T

. If G is a GB for

^a

w.r.t. a pure lexicographic term ordering ranking

^S

lower than

^T

then

^h

G

^\k



^S

]

ⁱ

=

^a\k



^S

] .

This statement is proved in e.g. 21] or 2].

Many functions in

^Polycon

use the Boege-Gebauer-Kredel (BGK) algorithm 1] for eliminating variables since this is in general more ecient than computing a GB w.r.t. a pure lexicographic term ordering. The BGK algorithm rst computes a GB which does not perform elimination, but which is comparatively cheap to compute. Then linear algebra techniques are used to nd the elimination ideal see 1, 17, 20] for more details. The BGK algorithm is implemented as

^finduni

in Maple. Most of the time, the BGK algorithm cannot be applied directly to the ideals considered in

^Polycon

. Instead the ideals must rst be localized so that they become zero-dimensional. This simply means that some of the variables are considered as parameters, i.e. the eld of coecients is extended to contain rational functions in some of the variables. Theoretical investigations of the localization procedure are presented in 16, 17].

The theoretical background, with proofs for the correctness of the algorithms in

^Polycon

, is given in chapters 5 and 6 of the author's PhD-thesis 12], where also some source code for primitive versions of the functions is given. Related theoretical work can be found in

5, 6, 19, 22, 23, 28, 33].

The theory behind

obsvc, obsvd, io2ssc

and

^io2ssd

is also described in 16] and 19], that of

^loclyap

in 11] and that of

^par2ssc

in 14].

4 An Example

Below is an example showing how some computations on a simple control system can be performed using

^Polycon

. The example is taken from 27], p. 147.

The following equations roughly describe the vertical motion of a hot air balloon:

_

x

1

=

^Ku;

1



(

^x1;T

) _

x

2

=

^x3

_

x

3

=

^{g Wa}

W

(1

^{; T}

x

1

)

^;

1

^{; }

W x

3



(6)

Here

^x2

is the altitude above sea level,

^x3

=

^{dtd x2}

and

^x1

is the air temperature (in Kelvin) inside the balloon. The constants

^{K Wa}

are related to the thermal and physical characteristics of the balloon,

^W

is the weight of the balloon including payload and air (in kg),

^T

is the outdoor temperature, and

^g

is the acceleration of gravity. The input

^u

of the system is the heat transfer rate supplied to the balloon by the heater. Let us consider the altitude as the output of the system:

^y

=

^x2

. If we put

^g

= 10

^W

= 200

^Wa

= 400 (I can't guarantee that these are realistic values

^:::

) we can use the procedure

^ss2ioc

in

^Polycon

to obtain a dierential equation relating the input and the output this equation is

8000

^Ty3;

400(

^T

+

^Ku

)

^y22;

40

^

(

^Ku

+

^T

)

^y1

+ 200

^Ku

+ 10

^Ty2

;

2

(

^Ku

+

^T

)

^y12

+ 400

^Kuy1

+ 40000(

^{T ;Ku}

) = 0 (7)

where

^y1

=

^dtdy

(

^t

),

^y2

=

^dtd22y

(

^t

) etc.

Now suppose that we have accurate measurements of the altitude

^x2

(so that we can compute the velocity and acceleration with acceptable precision) and want to estimate the temperature

^x1

in the balloon. We can then use the function

^obsvc

to nd an expression for

^x1

in the output and its derivatives. It turns out that

x

1

= 400

^T

200

^;

20

^y2;y1

(8)

(In fact, the observer relation is quite easy to derive by hand, in this case.) Below is a Maple session for doing the computations above:

> with(linalg): read(`polycon.m`):

> f:=vector(-(1/tau)*(x1-T)+K*u, x3, g*((Wa/W)*(1-(T/x1))-1-(mu/W)*x3)]):

> g:=10: Wa:=400: W:=200:

> p:=ss2ioc(f,x2):

> collect(p,y3,y2,y1])

8000 T tau y3 + (- 400 T - 400 K u0 tau) y22

+ ((- 40 mu K u0 tau - 40 mu T) y1 + 8000 K u0 tau + 400 mu T tau) y2

2 2 2

+ (- mu T - mu K u0 tau) y1 + 400 mu K u0 tau y1 + 40000 T - 40000 K u0 tau

> obsvc(f,x2,x1)

- 400 ---]T 20 y2 - 200 + y1 mu

5 Future Extensions

The current plans for future versions of

^Polycon

include:

Full support for rational functions. As mentioned in section 2 some functions fail on some rational systems. It is probably not dicult to amend this using the Rabinovich trick 7 , 13].

Identiability. It is possible to test global identiability of parameters in polynomial systems in state space form using GB. Some references, treating an even more general problem, are 24, 25, 35]. It seems straightforward to implement such functions in

Polycon

.

Implicit equations. Systems described by generalized state space equations, i.e. dif- ferential equations that are nonlinear in the derivatives can in some cases be handled within the GB framework.

Inequations. As pointed out in e.g. 5] and 33] it is sometimes necessary to add

inequations, i.e. expressions of the type

^q

(

^uy

)

⁶

= 0, to the input-output relation in

order for it to have the same solutions as the state equation. This is an important

aspect of the

^ss2io

-problem which is solved in the cited work, but still unclear in the

present framework.

Manifolds. Most of the

^Polycon

functions are possible to extend to systems having for state space a dierentiable manifold that is also an algebraic variety.

Time-varying systems.

As mentioned, there are many important parts of nonlinear control theory that are not covered by

^Polycon

, such as controllability and feedback. As the theoretical research con- tinues, algorithms for dealing with these problems will hopefully emerge and be implemented as far as possible.

Acknowledgement

This work was nancially supported by the Swedish Council for Technical Research (TFR).

References

1] W. Boege, R. Gebauer, and H. Kredel. Some examples for solving systems of algebraic equations by calculating Grobner bases. J. Symbolic Computation , 1:83{98, 1986.

2] B. Buchberger. Grobner bases: An algorithmic method in polynomial ideal theory. In N.K. Bose, editor, Multidimensional Systems Theory , pages 184{232. Dordrecht Reidel, 1985.

3] B. Char, K.O. Geddes, G.H. Gonnet, B.L. Leong, M.B. Monagan, and S.M. Watt.

Maple V Language Reference Manual . Springer, 1991.

4] B. de Jager. Symbolic calculation of zero dynamics for nonlinear control systems. In S.M. Watt, editor, Proc. ISSAC'91 , pages 321{322, Bonn, Germany, July 1991. ACM Press.

5] S. Diop. Elimination in control theory. Math. Control Signals Systems , 4(1):17{32, 1991.

6] S. Diop and M. Fliess. On nonlinear observability. In Proc. First European Control Conf. , volume 1, pages 152{157, Grenoble, France, July 1991. Herm"es.

7] A. Ferro and G. Gallo. Grobner bases, Ritt's algorithm and decision procedures for alge- braic theories. In L. Huguet and A. Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , volume 356 of LNCS , pages 230{237. Springer, 1989. Proc.

AAECC-5, Menorca.

8] M. Fliess. Some remarks on nonlinear invertibility and dynamic state-feedback. In C. Byrnes and A. Lindquist, editors, Theory and Applications of Nonlinear Control Systems , pages 115{121. North Holland, 1986.

9] M. Fliess. Automatique et corps di#erentiels. Forum Mathematicum , 1:227{238, 1989.

10] M. Fliess. Automatique en temps discret et alg"ebre aux di#erences. Forum Mathe-

maticum , 2:213{232, 1990.

11] K. Forsman. Applications of Grobner bases to nonlinear systems. In Proc. First European Control Conf. , volume 1, pages 164{169, Grenoble, France, July 1991. Herm"es.

12] K. Forsman. Constructive Commutative Algebra in Nonlinear Control Theory . PhD the- sis, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, 1991.

13] K. Forsman. Elementary aspects of constructive commutative algebra. Technical Re- port LiTH-ISY-I-1395, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, September 1992. Available by anonymous ftp

130.236.24.1

.

14] K. Forsman. On rational state space realizations. In M. Fliess, editor, Proc. NOLCOS'92 , pages 197{202, Bordeaux, 1992. IFAC.

15] K. Forsman.

^Polycon

- a Maple package for polynomial and rational control systems.

Technical Report LiTH-ISY-I-1386, Dept. of Electrical Engineering, Linkoping Univer- sity, S-581 83 Linkoping, Sweden, August 1992. Included in the Maple share library.

Also available by anonymous ftp

130.236.24.1

.

16] K. Forsman. Some generic results on algebraic observability and connections with re- alization theory. Technical Report LiTH-ISY-I-1403, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, September 1992. Available by anony- mous ftp

130.236.24.1

. Accepted for ECC '93.

17] K. Forsman. Localization and base change techniques in computational algebra. Tech- nical Report LiTH-ISY-R-1445, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, February 1993. Available by anonymous ftp

130.236.24.1

.

18] K.O. Geddes, S.R. Czapor, and G. Labahn. Algorithms for Computer Algebra . Kluwer Academic Publishers, 1992.

19] R. Germundsson and K. Forsman. A constructive approach to algebraic observability.

In Proc. 30:th CDC , volume 1, pages 451{452, Brighton, UK, 1991. IEEE.

20] P. Gianni and T. Mora. Algebraic solution of systems of polynomial equations using Grobner bases. In L. Huguet and A. Poli, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes , volume 356 of LNCS , pages 247{257. Springer, 1989. Proc.

AAECC-5, Menorca.

21] P. Gianni, B. Trager, and G. Zacharias. Grobner bases and primary decomposition of polynomial ideals. In L. Robbiano, editor, Computational Aspects of Commutative Algebra , pages 15{33. Academic Press, 1989. From J. Symb. Comp. Vol. 6, nr. 2-3.

22] S.T. Glad. Nonlinear state space and input output descriptions using dierential poly- nomials. In J. Descusse, M. Fliess, A. Isidori, and D. Leborgne, editors, New Trends in Nonlinear Control Theory , pages 182{189. Springer, 1988.

23] S.T. Glad. Dierential algebraic modelling of nonlinear systems. In M.A. Kaashoek, J.H.

van Schuppen, and A.C.M. Ran, editors, Realization and Modelling in System Theory.

Proc. Intl. Symp. MTNS-89 , volume I, pages 97{105, Amsterdam, 1990. Birkhauser.

24] S.T. Glad and L. Ljung. Model structure identiability and persistence of excitation. In

Proc. 29th CDC , volume 6, pages 3236{3240, Honolulu, Hawaii, 1990. IEEE.

25] S.T. Glad and L. Ljung. Parametrization of nonlinear model structures as linear regres- sions. In Proc. 11th IFAC World Congress , volume 6, pages 67{71, Tallinn, USSR, 1990.

IFAC.

26] T. Glad. Implementing Ritt's algorithm of dierential algebra. In M. Fliess, editor, Proc. NOLCOS'92 , pages 610{614, Bordeaux, 1992. IFAC.

27] N.H. McClamroch. State Models of Dynamic Systems - A Case Study Approach . Springer, 1980.

28] F. Ollivier. Le probl eme de l'identiabilite structurelle globale: approche theorique, methodes eectives et bornes de complexite . PhD thesis, #Ecole Polytechnique, 1990.

29] F. Pauer and M. Pfeifhofer. The theory of Grobner bases. L'Enseignement Mathe- matique , 34:215{232, 1988.

30] J.F. Ritt. Dierential Algebra. Dover, 1950.

31] E.D. Sontag. Polynomial Response Maps , volume 13 of Lecture Notes in Control and Information Sciences . Springer, 1979.

32] T. Svensson. Mathematical Tools and Software for Analysis and Design of Nonlinear Control Systems . PhD thesis, Dept. of Electrical Engineering, Linkoping University, S-581 83 Linkoping, Sweden, 1992.

33] A.J. van der Schaft. Representing a nonlinear state space system as a set of higher-order dierential equations in the inputs and outputs. Systems & Control Letters , 12(2):151{

160, February 1989.

34] H. van Essen. Symbols speak louder than numbers: Analysis and design of nonlinear control systems with the symbolic computation system Maple. Master's thesis, Dept. of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Nether- lands, June 1992. Number WFW 92.061.

35] E. Walter, Y. Lecourtier, and A. Raksanyi. Test of structural properties of state space

models through algebraic computation. In Proc. 9th IFAC World Congress , volume 8,

pages 250{255, Budapest, 1984. IFAC.